Given that $\begin{bmatrix} 3&7.5 \\-6 &4.5 \end{bmatrix}_{2\times 2}.\begin{bmatrix}x \\y \end{bmatrix}_{2\times 1}=\begin{bmatrix}6 \\-90\end{bmatrix}_{2\times 1}$
We can simply multiply and get
$\begin{bmatrix} 3x+7.5y\\-6x+4.5y \end{bmatrix}_{2\times 1}=\begin{bmatrix}6 \\-90\end{bmatrix}_{2\times 1}$
compare and write
$3x+7.5y=6$
multiply by $10$ on both sides, we get
$30x+75y=60$
divide by $15$ into both sides, we get
$2x+5y=4$----------$>(1)$
and $-6x+4.5y=-90$
multiply by $10$ on both sides, we get
$-60x+45y-900$
divide by $15$ into both sides, we get
$-4x+3y=-60$-------$>(2)$
from equation $(1)$ and $(2)$,we can solve and get $x=12$ and $y=-4$